# K\"ahler differential algebras for 0-dimensional schemes

## TL;DR

This paper investigates the structure of Kähler differential algebras associated with 0-dimensional schemes in projective space, providing explicit calculations and bounds for their Hilbert functions and polynomials.

## Contribution

It offers explicit presentations and detailed analysis of Kähler differential modules for 0-dimensional schemes, including bounds and specific cases like subschemes of P^1 and P^2.

## Key findings

- Explicit Hilbert function values for differential modules
- Bounds on Hilbert polynomials and regularity indices
- Detailed results for schemes on a conic in P^2

## Abstract

Given a 0-dimensional scheme in a projective space $\mathbb{P}^n$ over a field $K$, we study the K\"ahler differential algebra $\Omega_{R/K}$ of its homogeneous coordinate ring $R$. Using explicit presentations of the modules $\Omega^m_{R/K}$ of K\"ahler differential $m$-forms, we determine many values of their Hilbert functions explicitly and bound their Hilbert polynomials and regularity indices. Detailed results are obtained for subschemes of $\mathbb{P}^1$, fat point schemes, and subschemes of $\mathbb{P}^2$ supported on a conic.